
# Rose wine sales


import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns


df_1=pd.read_csv(r'C:\Users\Sandeep Immadi\Downloads\Rose.csv')
df_1.head()


df_2 = pd.read_csv(r'C:\Users\Sandeep Immadi\Downloads\Rose.csv',parse_dates=True,squeeze=True,index_col=0)


print(df_2.head())
print('\n')
print(df_2.tail())

YearMonth
1980-01-01    112.0
1980-02-01    118.0
1980-03-01    129.0
1980-04-01     99.0
1980-05-01    116.0
Name: Rose, dtype: float64


YearMonth
1995-03-01    45.0
1995-04-01    52.0
1995-05-01    28.0
1995-06-01    40.0
1995-07-01    62.0
Name: Rose, dtype: float64


date=pd.date_range(start='1/1/1980',end='8/1/1995',freq='M')
date

DatetimeIndex(['1980-01-31', '1980-02-29', '1980-03-31', '1980-04-30',
               '1980-05-31', '1980-06-30', '1980-07-31', '1980-08-31',
               '1980-09-30', '1980-10-31',
               ...
               '1994-10-31', '1994-11-30', '1994-12-31', '1995-01-31',
               '1995-02-28', '1995-03-31', '1995-04-30', '1995-05-31',
               '1995-06-30', '1995-07-31'],
              dtype='datetime64[ns]', length=187, freq='M')


df_1['Time_Stamp'] = pd.DataFrame(date,columns=['Month'])
df_1.head()


df = df_1.set_index('Time_Stamp',)
df.drop('YearMonth',axis=1,inplace=True)
df.head()


# Plotting the time series for 'Rose' wine sales


df.plot(figsize=(20,8))
plt.grid()
plt.title("Rose Sales")
plt.show()


# Checking null values for 'Rose' wine sales
df.isnull().sum()

Rose    2
dtype: int64


df=df.interpolate(method='linear',limit_direction='forward')


df.isnull().sum()

Rose    0
dtype: int64


# Descriptive Analysis of 'Sparkling' Wine Sales
df.describe()


# Plotting Boxplot for year wise
from pylab import rcParams
rcParams['figure.figsize'] = 20,8


sns.boxplot(x=df.index.year,y=df['Rose'])
plt.grid();


# Plotting Boxplot for Month wise

sns.boxplot(x=df.index.month_name(),y=df['Rose'])
plt.grid();


# Plotting the Rose wine year month wise sales - Line plot


monthly_sales_across_years=pd.pivot_table(df,values='Rose',columns=df.index.month_name(),index=df.index.year)
monthly_sales_across_years


monthly_sales_across_years.plot()
plt.grid()
plt.legend(loc='best');


# Plotting a month plot to check the sales in different years and withing different month across


## importing the month plot from statsmodels
from statsmodels.graphics.tsaplots import month_plot


month_plot(df['Rose'],ylabel='Rose Sales')
plt.grid();


# Plot the Empirical Cumulative Distribution


# statistics
from statsmodels.distributions.empirical_distribution import ECDF
plt.figure(figsize = (18, 8))
cdf = ECDF(df['Rose'])
plt.plot(cdf.x, cdf.y, label = "statmodels");
plt.grid()
plt.xlabel('Sales');


# Plot the average Rose Wine Sales per month and the month on month percentage change of Rose Wine Sales


# group by date and get average Rose wine Sales, and precent change
average = df.groupby(df.index)["Rose"].mean()
pct_change = df.groupby(df.index)["Rose"].sum().pct_change()
fig, (axis1,axis2) = plt.subplots(2,1,sharex=True,figsize=(15,8))
# plot average Rose wine Sales over time(year-month)
ax1 = average.plot(legend=True,ax=axis1,marker='o',title="Average Rose Wine Sales",grid=True)
ax1.set_xticks(range(len(average)))
ax1.set_xticklabels(average.index.tolist())
# plot precent change for Rpse Wine Sales over time(year-month)
ax2 = pct_change.plot(legend=True,ax=axis2,marker='o',colormap="summer",title="Rose wine Sales Percent Change",grid=True)


# Decomposing the time series into Additive decomposition and plot


from statsmodels.tsa.seasonal import seasonal_decompose
decomposition = seasonal_decompose(df['Rose'],model='Additive')
decomposition.plot();


trend= decomposition.trend
seasonality=decomposition.seasonal
residual=decomposition.resid
print('Trend','\n',trend.head(12),'\n')
print('Seasonality','\n',seasonality.head(12),'\n')
print('Residual','\n',residual.head(12),'\n')

Trend 
 Time_Stamp
1980-01-31           NaN
1980-02-29           NaN
1980-03-31           NaN
1980-04-30           NaN
1980-05-31           NaN
1980-06-30           NaN
1980-07-31    147.083333
1980-08-31    148.125000
1980-09-30    148.375000
1980-10-31    148.083333
1980-11-30    147.416667
1980-12-31    145.125000
Name: trend, dtype: float64 

Seasonality 
 Time_Stamp
1980-01-31   -27.908647
1980-02-29   -17.435632
1980-03-31    -9.285830
1980-04-30   -15.098330
1980-05-31   -10.196544
1980-06-30    -7.678687
1980-07-31     4.896908
1980-08-31     5.499686
1980-09-30     2.774686
1980-10-31     1.871908
1980-11-30    16.846908
1980-12-31    55.713575
Name: seasonal, dtype: float64 

Residual 
 Time_Stamp
1980-01-31          NaN
1980-02-29          NaN
1980-03-31          NaN
1980-04-30          NaN
1980-05-31          NaN
1980-06-30          NaN
1980-07-31   -33.980241
1980-08-31   -24.624686
1980-09-30    53.850314
1980-10-31    -2.955241
1980-11-30   -14.263575
1980-12-31    66.161425
Name: resid, dtype: float64


#Decomposing the time series into Additive decomosition and plot


decomposition = seasonal_decompose(df['Rose'],model='multiplicative')
decomposition.plot();


trend= decomposition.trend
seasonality=decomposition.seasonal
residual=decomposition.resid
print('Trend','\n',trend.head(12),'\n')
print('Seasonality','\n',seasonality.head(12),'\n')
print('Residual','\n',residual.head(12),'\n')

Trend 
 Time_Stamp
1980-01-31           NaN
1980-02-29           NaN
1980-03-31           NaN
1980-04-30           NaN
1980-05-31           NaN
1980-06-30           NaN
1980-07-31    147.083333
1980-08-31    148.125000
1980-09-30    148.375000
1980-10-31    148.083333
1980-11-30    147.416667
1980-12-31    145.125000
Name: trend, dtype: float64 

Seasonality 
 Time_Stamp
1980-01-31    0.670111
1980-02-29    0.806163
1980-03-31    0.901164
1980-04-30    0.854024
1980-05-31    0.889415
1980-06-30    0.923985
1980-07-31    1.058038
1980-08-31    1.035881
1980-09-30    1.017648
1980-10-31    1.022573
1980-11-30    1.192349
1980-12-31    1.628646
Name: seasonal, dtype: float64 

Residual 
 Time_Stamp
1980-01-31         NaN
1980-02-29         NaN
1980-03-31         NaN
1980-04-30         NaN
1980-05-31         NaN
1980-06-30         NaN
1980-07-31    0.758258
1980-08-31    0.840720
1980-09-30    1.357674
1980-10-31    0.970771
1980-11-30    0.853378
1980-12-31    1.129646
Name: resid, dtype: float64


# Splitting the data into train and test data


train=df[df.index.year < 1991]
test=df[df.index.year >= 1991]


## This is to display multiple data frames from one cell
from IPython.display import display


print('First few rows of Training Data')
display(train.head())
print('Last few rows of Training Data')
display(train.tail())
print('First few rows of Test Data')
display(test.head())
print('Last few rows of Test Data')
display(test.tail())

First few rows of Training Data

Last few rows of Training Data

First few rows of Test Data

Last few rows of Test Data


# Checking the shape of the train and test data
print(train.shape)
print(test.shape)

(132, 1)
(55, 1)


# plotting the graph for train and test set

train['Rose'].plot(fontsize=14)
test['Rose'].plot(fontsize=14)
plt.grid()
plt.legend(['Training data','Test data'])
plt.show()


train_time=[i+1 for i in range(len(train))]
test_time=[i+133 for i in range(len(test))]
print('Training Time instance','\n',train_time)
print('\n')
print('Test Time instance','\n',test_time)

Training Time instance 
 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132]


Test Time instance 
 [133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187]


LinearRegression_train = train.copy()
LinearRegression_test = test.copy()


LinearRegression_train['time'] = train_time
LinearRegression_test['time'] = test_time
print('First few rows of Training Data','\n',LinearRegression_train.head(),'\n')
print('Last few rows of Training Data','\n',LinearRegression_train.tail(),'\n')
print('First few rows of Test Data','\n',LinearRegression_test.head(),'\n')
print('Last few rows of Test Data','\n',LinearRegression_test.tail(),'\n')

First few rows of Training Data 
              Rose  time
Time_Stamp             
1980-01-31  112.0     1
1980-02-29  118.0     2
1980-03-31  129.0     3
1980-04-30   99.0     4
1980-05-31  116.0     5 

Last few rows of Training Data 
              Rose  time
Time_Stamp             
1990-08-31   70.0   128
1990-09-30   83.0   129
1990-10-31   65.0   130
1990-11-30  110.0   131
1990-12-31  132.0   132 

First few rows of Test Data 
             Rose  time
Time_Stamp            
1991-01-31  54.0   133
1991-02-28  55.0   134
1991-03-31  66.0   135
1991-04-30  65.0   136
1991-05-31  60.0   137 

Last few rows of Test Data 
             Rose  time
Time_Stamp            
1995-03-31  45.0   183
1995-04-30  52.0   184
1995-05-31  28.0   185
1995-06-30  40.0   186
1995-07-31  62.0   187


from sklearn.linear_model import LinearRegression


lr = LinearRegression()


lr.fit(LinearRegression_train[['time']],LinearRegression_train['Rose'].values)

LinearRegression()


# Plotting Linear Regression Forecast
train_predictions_model1= lr.predict(LinearRegression_train[['time']])
LinearRegression_train['RegOnTime'] = train_predictions_model1
test_predictions_model1= lr.predict(LinearRegression_test[['time']])
LinearRegression_test['RegOnTime']=test_predictions_model1
plt.plot( train['Rose'], label='Train')
plt.plot(test['Rose'], label='Test')
plt.plot(LinearRegression_train['RegOnTime'],label='Regression On Time_Train_Data')
plt.plot(LinearRegression_test['RegOnTime'], label='Regression On Time_Test Data')
plt.legend(loc='best')
plt.grid();


# Model 1 Evaluation:


### Defining Accuacy Matrix]
from sklearn import metrics
from math import sqrt


# Train - RMSE score


rmse_model1_train = sqrt(metrics.mean_squared_error(train['Rose'],train_predictions_model1))
print("For RegressionOnTime forecast on the Train Data, RMSE is %3.3f" %(rmse_model1_train))

For RegressionOnTime forecast on the Train Data, RMSE is 30.718


# Test - RMSE score


rmse_model1_test = sqrt(metrics.mean_squared_error(test['Rose'],test_predictions_model1))
print("For RegressionOnTime forecast on the Test Data, RMSE is %3.3f" %(rmse_model1_test))

For RegressionOnTime forecast on the Test Data, RMSE is 15.269


# Creating Dataframe


result_DF=pd.DataFrame({'Train_RMSE':[rmse_model1_train],'Test_RMSE':[rmse_model1_test]},index=['RegressionOnTime'])
result_DF


NaiveModel_train = train.copy()
NaiveModel_test = test.copy()


NaiveModel_train['naive'] = np.asarray(train['Rose'])[len(np.asarray(train['Rose']))-1]
NaiveModel_train['naive'].head()

Time_Stamp
1980-01-31    132.0
1980-02-29    132.0
1980-03-31    132.0
1980-04-30    132.0
1980-05-31    132.0
Name: naive, dtype: float64


NaiveModel_test['naive'] = np.asarray(train['Rose'])[len(np.asarray(train['Rose']))-1]
NaiveModel_test['naive'].head()

Time_Stamp
1991-01-31    132.0
1991-02-28    132.0
1991-03-31    132.0
1991-04-30    132.0
1991-05-31    132.0
Name: naive, dtype: float64


# Plotting Naive Forecast


plt.plot( train['Rose'], label='Train')
plt.plot(test['Rose'], label='Test')
plt.plot(NaiveModel_train['naive'],label='Naive Forecast On Time_Train_Data')
plt.plot(NaiveModel_test['naive'], label='Naive forecast On Time_Test Data')
plt.legend(loc='best')
plt.title('Naive Forecast')
plt.grid();


# Model 2 Evaluation


# Train - RMSE score
rmse_model2_train = sqrt(metrics.mean_squared_error(train['Rose'],NaiveModel_train['naive']))
print("For Naive Model forecast on the Train Data, RMSE is %3.3f" %(rmse_model2_train))

For Naive Model forecast on the Train Data, RMSE is 45.064


# Test-RMSE score
rmse_model2_test = sqrt(metrics.mean_squared_error(test['Rose'],NaiveModel_test['naive']))
print("For Naive Model forecast on the Test Data, RMSE is %3.3f" %(rmse_model2_test))

For Naive Model forecast on the Test Data, RMSE is 79.719


# Creating Dataframe


result_DF_2=pd.DataFrame({'Train_RMSE':[rmse_model2_train],'Test_RMSE':[rmse_model2_test]},index=['Naive Model'])
result_DF=pd.concat([result_DF,result_DF_2])
result_DF


SimpleAverage_Train=train.copy()
SimpleAverage_Test=test.copy()


SimpleAverage_Train['mean_forecast']= train['Rose'].mean()
SimpleAverage_Train.head()


SimpleAverage_Test['mean_forecast']= train['Rose'].mean()
SimpleAverage_Test.head()


# Plotting Simple Average Forecast


plt.plot(SimpleAverage_Train['Rose'], label='Train')
plt.plot(SimpleAverage_Test['Rose'], label='Test')
plt.plot(SimpleAverage_Train['mean_forecast'],label='Simple Average on Training Data')
plt.plot(SimpleAverage_Test['mean_forecast'], label='Simple Average on Test Data')
plt.legend(loc='best')
plt.title("Simple Average Forecast")
plt.grid();


# Model 3 Evaluation


# Train - RMSE score
rmse_model3_train = sqrt(metrics.mean_squared_error(train['Rose'],SimpleAverage_Train['mean_forecast']))
print("For Simple Average forecast on the Train Data, RMSE is %3.3f" %(rmse_model3_train))

For Simple Average forecast on the Train Data, RMSE is 36.034


# Test - RMSE score
rmse_model3_test = sqrt(metrics.mean_squared_error(test['Rose'],SimpleAverage_Test['mean_forecast']))
print("For Simple Average forecast on the Test Data, RMSE is %3.3f" %(rmse_model3_test))

For Simple Average forecast on the Test Data, RMSE is 53.461


# Creating DataFrame


result_DF_3=pd.DataFrame({'Train_RMSE':[rmse_model3_train],'Test_RMSE':[rmse_model3_test]},index=['Simple Average'])
result_DF=pd.concat([result_DF,result_DF_3])
result_DF


MovingAverage = df.copy()
MovingAverage.head()


# Trailing Moving Averages
MovingAverage['Trailing_2'] = MovingAverage['Rose'].rolling(2).mean()
MovingAverage['Trailing_4'] = MovingAverage['Rose'].rolling(4).mean()
MovingAverage['Trailing_6'] = MovingAverage['Rose'].rolling(6).mean()
MovingAverage['Trailing_9'] = MovingAverage['Rose'].rolling(9).mean()
MovingAverage.head()


# Plotting Moving Average Forecast


plt.plot(MovingAverage['Rose'], label='Train')
plt.plot(MovingAverage['Trailing_2'], label='2 Point Moving Average')
plt.plot(MovingAverage['Trailing_4'], label='4 Point Moving Average')
plt.plot(MovingAverage['Trailing_6'],label = '6 Point Moving Average')
plt.plot(MovingAverage['Trailing_9'],label = '9 Point Moving Average')
plt.legend(loc = 'best')
plt.grid();


# Creating Train and Test set

trailing_MovingAverage_train=MovingAverage[0:int(len(MovingAverage)*0.71)]
trailing_MovingAverage_test=MovingAverage[int(len(MovingAverage)*0.71):]


# Plotting Grpah for Trailing MA


plt.plot(trailing_MovingAverage_train['Rose'], label='Train')
plt.plot(trailing_MovingAverage_test['Rose'], label='Test')
plt.plot(trailing_MovingAverage_train['Trailing_2'], label='2 Point Trailing Moving Average on Training Set')
plt.plot(trailing_MovingAverage_train['Trailing_4'], label='4 Point Trailing Moving Average on Training Set')
plt.plot(trailing_MovingAverage_train['Trailing_6'],label = '6 Point Trailing Moving Average on Training Set')
plt.plot(trailing_MovingAverage_train['Trailing_9'],label = '9 Point Trailing Moving Average on Training Set')
plt.plot(trailing_MovingAverage_test['Trailing_2'], label='2 Point Trailing Moving Average on Test Set')
plt.plot(trailing_MovingAverage_test['Trailing_4'], label='4 Point Trailing Moving Average on Test Set')
plt.plot(trailing_MovingAverage_test['Trailing_6'],label = '6 Point Trailing Moving Average on Test Set')
plt.plot(trailing_MovingAverage_test['Trailing_9'],label = '9 Point Trailing Moving Average on Test Set')
plt.legend(loc = 'best')
plt.grid();


# RMSE score for Trailing MA


## Test Data - RMSE --> 2 point Trailing MA
rmse_model4_test_2 = sqrt(metrics.mean_squared_error(test['Rose'],trailing_MovingAverage_test['Trailing_2']))
print("For 2 point Moving Average Model forecast on the Training Data, RMSE is %3.3f" %(rmse_model4_test_2))
## Test Data - RMSE --> 4 point Trailing MA
rmse_model4_test_4 = sqrt(metrics.mean_squared_error(test['Rose'],trailing_MovingAverage_test['Trailing_4']))
print("For 4 point Moving Average Model forecast on the Training Data, RMSE is %3.3f" %(rmse_model4_test_4))
## Test Data - RMSE --> 6 point Trailing MA
rmse_model4_test_6 = sqrt(metrics.mean_squared_error(test['Rose'],trailing_MovingAverage_test['Trailing_6']))
print("For 6 point Moving Average Model forecast on the Training Data, RMSE is %3.3f" %(rmse_model4_test_6))
## Test Data - RMSE --> 9 point Trailing MA
rmse_model4_test_9 = sqrt(metrics.mean_squared_error(test['Rose'],trailing_MovingAverage_test['Trailing_9']))
print("For 9 point Moving Average Model forecast on the Training Data, RMSE is %3.3f " %(rmse_model4_test_9))

For 2 point Moving Average Model forecast on the Training Data, RMSE is 11.529
For 4 point Moving Average Model forecast on the Training Data, RMSE is 14.451
For 6 point Moving Average Model forecast on the Training Data, RMSE is 14.566
For 9 point Moving Average Model forecast on the Training Data, RMSE is 14.728


# Creating Dataframe


resultsDf_4 = pd.DataFrame({'Test_RMSE': [rmse_model4_test_2,rmse_model4_test_4
 ,rmse_model4_test_6,rmse_model4_test_9]}
 ,index=['2pointTrailingMovingAverage','4pointTrailingMovingAverage'
 ,'6pointTrailingMovingAverage','9pointTrailingMovingAverage'])
result_DF = pd.concat([result_DF, resultsDf_4])
result_DF


## Importing Simple exponential Smoothing 
from statsmodels.tsa.holtwinters import ExponentialSmoothing, SimpleExpSmoothing, Holt


SES_Train=train.copy()
SES_test=test.copy()


model_SES=SimpleExpSmoothing(SES_Train['Rose'],)

C:\ProgramData\Anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency M will be used.
  warnings.warn('No frequency information was'
C:\ProgramData\Anaconda3\lib\site-packages\statsmodels\tsa\holtwinters\model.py:427: FutureWarning: After 0.13 initialization must be handled at model creation
  warnings.warn(


model_SES_autofit=model_SES.fit(optimized=True,smoothing_level=0.995)


model_SES_autofit.params

{'smoothing_level': 0.995,
 'smoothing_trend': nan,
 'smoothing_seasonal': nan,
 'damping_trend': nan,
 'initial_level': 112.03027126088527,
 'initial_trend': nan,
 'initial_seasons': array([], dtype=float64),
 'use_boxcox': False,
 'lamda': None,
 'remove_bias': False}


# Predicting for trainig data


SES_Train['predict'] = model_SES_autofit.fittedvalues
SES_Train.head()


# Predicting for Test data


SES_test['predict'] = model_SES_autofit.forecast(steps=55)
SES_test.head()


# Plotting graph for Simple Exponential Smooting forecast

plt.plot(SES_Train['Rose'], label='Train')
plt.plot(SES_test['Rose'], label='Test')
plt.plot(SES_Train['predict'],label='Alpha = 0.995 Simple Exponential Smoothing Predictions on Training set')
plt.plot(SES_test['predict'], label='Alpha = 0.995 Simple Exponential Smoothing predictions on Test Set')
plt.legend(loc='best')
plt.grid()
plt.title('Alpha = 0.995 Predictions');


# Train - RMSE score


rmse_model5_Train_1 = metrics.mean_squared_error(SES_Train['Rose'],SES_Train['predict'],squared=False)
print("For Alpha =0.995 Simple Exponential Smoothing Model forecast on the Train Data, RMSE is %3.3f" %(rmse_model5_Train_1))

For Alpha =0.995 Simple Exponential Smoothing Model forecast on the Train Data, RMSE is 38.715


# Test - RMSE score


rmse_model5_Test_1 = metrics.mean_squared_error(SES_test['Rose'],SES_test['predict'],squared=False)
print("For Alpha =0.995 Simple Exponential Smoothing Model forecast on the Test Data, RMSE is %3.3f" %(rmse_model5_Test_1))

For Alpha =0.995 Simple Exponential Smoothing Model forecast on the Test Data, RMSE is 79.610


# Creating DataFrame


resultsDf_5 = pd.DataFrame({'Train_RMSE' : [rmse_model5_Train_1],
 'Test_RMSE': [rmse_model5_Test_1]},index=['Alpha=0.995,SimpleExponentialSmoothing'])
result_DF = pd.concat([result_DF, resultsDf_5])
result_DF


# Setting Different Alpha Values
resultsDf_6 = pd.DataFrame({'Alpha_Values':[],'Train_RMSE':[],'Test_RMSE': []})
resultsDf_6


for i in np.arange(0.3,1,0.1):
 model_SES_alpha_i = model_SES.fit(smoothing_level=i,optimized=False,use_brute=True)
 SES_Train['predict',i] = model_SES_alpha_i.fittedvalues
 SES_test['predict',i] = model_SES_alpha_i.forecast(steps=55)
 
 rmse_model5_train_i = metrics.mean_squared_error(SES_Train['Rose'],SES_Train['predict',i],squared=False)
 
 rmse_model5_test_i = metrics.mean_squared_error(SES_test['Rose'],SES_test['predict',i],squared=False)
 
 resultsDf_6 = resultsDf_6.append({'Alpha_Values':i,'Train_RMSE':rmse_model5_train_i
 ,'Test_RMSE':rmse_model5_test_i}, ignore_index=True)


# Model Evaluation


resultsDf_6.sort_values(by=['Test_RMSE'],ascending=True)


# Plotting graph for different Alpha values of Traninig and Test data


plt.figure(figsize=(18,9))
plt.plot(SES_Train['Rose'], label='Train')
plt.plot(SES_test['Rose'], label='Test')
plt.plot(SES_test['predict'], label='Alpha = 1 Simple Exponential Smoothing predictions on Test Set')
plt.plot(SES_test['predict', 0.3], label='Alpha =0.3 Simple Exponential Smoothing predictions on Test Set')
plt.legend(loc='best')
plt.grid();


# Creating Dataframe


resultsDf_6_1 = pd.DataFrame({'Train_RMSE' : [resultsDf_6.sort_values(by=['Train_RMSE'],ascending=True).values[0][1]],
                              'Test_RMSE': [resultsDf_6.sort_values(by=['Test_RMSE'],ascending=True).values[0][2]]}
 ,index=['Alpha=0.3,SimpleExponentialSmoothing'])
result_DF = pd.concat([result_DF, resultsDf_6_1])
result_DF


# Model 6 : Double Exponential Smoothing

DES_train = train.copy()
DES_test = test.copy()


model_DES = Holt(DES_train['Rose'])

C:\ProgramData\Anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:524: ValueWarning: No frequency information was provided, so inferred frequency M will be used.
  warnings.warn('No frequency information was'
C:\ProgramData\Anaconda3\lib\site-packages\statsmodels\tsa\holtwinters\model.py:427: FutureWarning: After 0.13 initialization must be handled at model creation
  warnings.warn(


## First we will define an empty dataframe to store our values from the loop
resultsDf_7 = pd.DataFrame({'Alpha_Values':[],'Beta_Values':[],'Train_RMSE':[],'Test_RMSE': []})
resultsDf_7


for i in np.arange(0.3,1.1,0.1):
    for j in np.arange(0.3,1.1,0.1):
        model_DES_alpha_i_j = model_DES.fit(smoothing_level=i,smoothing_trend=j,optimized=False,use_brute=True)
        DES_train['predict',i,j] = model_DES_alpha_i_j.fittedvalues
        DES_test['predict',i,j] = model_DES_alpha_i_j.forecast(steps=55)
        
        rmse_model6_train = metrics.mean_squared_error(DES_train['Rose'],DES_train['predict',i,j],squared=False)
        
        rmse_model6_test = metrics.mean_squared_error(DES_test['Rose'],DES_test['predict',i,j],squared=False)
        
        resultsDf_7 = resultsDf_7.append({'Alpha_Values':i,'Beta_Values':j,'Train_RMSE':rmse_model6_train,'Test_RMSE':rmse_model6_test}, ignore_index=True)


resultsDf_7


resultsDf_7.sort_values(by=['Test_RMSE']).head()


# Plotting Graph on Both Traning and Test data


plt.figure(figsize=(18,9))
plt.plot(DES_train['Rose'], label='Train')
plt.plot(DES_test['Rose'], label='Test')
plt.plot(DES_train['predict', 0.3, 0.3], label='Alpha=0.3,Beta=0.3,DoubleExponentialSmoothing predictions on Training set')
plt.plot(DES_test['predict', 0.3, 0.3], label='Alpha=0.3,Beta=0.3,DoubleExponentialSmoothing predictions on Test Set')
plt.legend(loc='best')
plt.grid();


resultsDf_7_1 = pd.DataFrame({'Train_RMSE' :[resultsDf_7.sort_values(by=['Train_RMSE']).values[0][2]],
 'Test_RMSE': [resultsDf_7.sort_values(by=['Test_RMSE']).values[0][3]]}
 ,index=['Alpha=0.3,Beta=0.3,DoubleExponentialSmoothing'])
result_DF = pd.concat([result_DF, resultsDf_7_1])
result_DF


TES_train = train.copy()
TES_test = test.copy()


model_TES = ExponentialSmoothing(TES_train['Rose'],trend='additive',seasonal='multiplicative',freq='M')

C:\ProgramData\Anaconda3\lib\site-packages\statsmodels\tsa\holtwinters\model.py:427: FutureWarning: After 0.13 initialization must be handled at model creation
  warnings.warn(


model_TES_autofit = model_TES.fit()


model_TES_autofit.params

{'smoothing_level': 0.06571007449183297,
 'smoothing_trend': 0.051867105713176015,
 'smoothing_seasonal': 0.0015637515713898,
 'damping_trend': nan,
 'initial_level': 47.81887301367471,
 'initial_trend': -0.2961562797665537,
 'initial_seasons': array([2.35763018, 2.67367218, 2.92146068, 2.55308191, 2.87099548,
        3.13124987, 3.44178442, 3.66118656, 3.47154364, 3.39670325,
        3.95879831, 5.46173463]),
 'use_boxcox': False,
 'lamda': None,
 'remove_bias': False}


# Predicton on Train Set


TES_train['auto_predict'] = model_TES_autofit.fittedvalues
TES_train.head()


# Prediction on Test set


TES_test['auto_predict'] = model_TES_autofit.forecast(steps=len(test))
TES_test.head()


# Plotting graph using Training set and Test set using autofit


plt.plot(TES_train['Rose'], label='Train')
plt.plot(TES_test['Rose'], label='Test')
plt.plot(TES_train['auto_predict'], label='Alpha=0.064,Beta=0.053,Gamma=0.0,TripleExponentialSmoothing predictions on Test Set')
plt.plot(TES_test['auto_predict'], label='Alpha=0.064,Beta=0.053,Gamma=0.0,TripleExponentialSmoothing predictions on Test Set') 
plt.legend(loc='best')
plt.grid();


# Train - RMSE score


rmse_model6_train_1 = metrics.mean_squared_error(TES_train['Rose'],TES_train['auto_predict'],squared=False)
print("For Alpha=0.064,Beta=0.053,Gamma=0.0, Triple Exponential Smoothing Model forecast on the Train Data, RMSE is %3.3f" %(rmse_model6_train_1))

For Alpha=0.064,Beta=0.053,Gamma=0.0, Triple Exponential Smoothing Model forecast on the Train Data, RMSE is 18.415


rmse_model6_test_1 = metrics.mean_squared_error(TES_test['Rose'],TES_test['auto_predict'],squared=False)
print("For Alpha=0.064,Beta=0.053,Gamma=0.0, Triple Exponential Smoothing Model forecast on the Test Data, RMSE is %3.3f" %(rmse_model6_test_1))

For Alpha=0.064,Beta=0.053,Gamma=0.0, Triple Exponential Smoothing Model forecast on the Test Data, RMSE is 20.990


# Creating DataFrame


resultsDf_8_1 = pd.DataFrame({'Train_RMSE':[rmse_model6_train_1],
 'Test_RMSE': [rmse_model6_test_1]}
 ,index=['Alpha=0.064,Beta=0.053,Gamma=0.0,TripleExponentialSmoothing'])
result_DF = pd.concat([result_DF, resultsDf_8_1])
result_DF


## First we will define an empty dataframe to store our values from the loop
resultsDf_8_2 = pd.DataFrame({'Alpha_Values':[],'Beta_Values':[],'Gamma_Values':[],'Train_RMSE':[],'Test_RMSE': []})
resultsDf_8_2


for i in np.arange(0.3,1.1,0.1):
    for j in np.arange(0.3,1.1,0.1):
         for k in np.arange(0.3,1.1,0.1):
                model_TES_alpha_i_j_k = model_TES.fit(smoothing_level=i,smoothing_trend=j,smoothing_seasonal=k,optimized=False,use_brute=True)
                TES_train['predict',i,j,k] = model_TES_alpha_i_j_k.fittedvalues
                TES_test['predict',i,j,k] = model_TES_alpha_i_j_k.forecast(steps=55)
                rmse_model8_train = metrics.mean_squared_error(TES_train['Rose'],TES_train['predict',i,j,k],squared=False)
                rmse_model8_test = metrics.mean_squared_error(TES_test['Rose'],TES_test['predict',i,j,k],squared=False)
                resultsDf_8_2 = resultsDf_8_2.append({'Alpha_Values':i,'Beta_Values':j,'Gamma_Values':k,
                                                          'Train_RMSE':rmse_model8_train,'Test_RMSE':rmse_model8_test}
                                                          , ignore_index=True)


resultsDf_8_2


resultsDf_8_2.sort_values(by=['Test_RMSE']).head()


## Plotting on both the Training and Test data using brute force alpha, beta and gamma determination
plt.plot(TES_train['Rose'], label='Train')
plt.plot(TES_test['Rose'], label='Test')
#The value of alpha and beta is taken like that by python
plt.plot(TES_train['predict', 0.3, 0.4, 0.3], label='Alpha=0.3,Beta=0.4,Gamma=0.3,TripleExponentialSmoothing prediction on Training set')
plt.plot(TES_test['predict', 0.3, 0.4, 0.3], label='Alpha=0.3,Beta=0.4,Gamma=0.3,TripleExponentialSmoothing prediction on Testing set')
plt.legend(loc='best')
plt.grid();


# Creating Dataframe


resultsDf_8_3 = pd.DataFrame({'Train_RMSE':[resultsDf_8_2.sort_values(by=['Test_RMSE']).values[0][3]],
 'Test_RMSE': [resultsDf_8_2.sort_values(by=['Test_RMSE']).values[0][4]]}
 ,index=['Alpha=0.3,Beta=0.4,Gamma=0.3,TripleExponentialSmoothing'])
result_DF = pd.concat([result_DF, resultsDf_8_3])
result_DF


print('Sorted by RMSE values on Test Data','\n',result_DF.sort_values(by=['Test_RMSE']),'\n','\n')
print('Sorted by RMSE values on Train Data','\n',result_DF.sort_values(by=['Train_RMSE']))

Sorted by RMSE values on Test Data 
                                                     Train_RMSE   Test_RMSE
Alpha=0.3,Beta=0.4,Gamma=0.3,TripleExponentialS...   28.111886   10.945435
2pointTrailingMovingAverage                                NaN   11.529278
4pointTrailingMovingAverage                                NaN   14.451403
6pointTrailingMovingAverage                                NaN   14.566327
9pointTrailingMovingAverage                                NaN   14.727630
RegressionOnTime                                     30.718135   15.268955
Alpha=0.064,Beta=0.053,Gamma=0.0,TripleExponent...   18.414568   20.990268
Alpha=0.3,SimpleExponentialSmoothing                 32.470164   47.504821
Simple Average                                       36.034234   53.460570
Alpha=0.995,SimpleExponentialSmoothing               38.714722   79.609847
Naive Model                                          45.063760   79.718773
Alpha=0.3,Beta=0.3,DoubleExponentialSmoothing        35.944983  265.567594 
 

Sorted by RMSE values on Train Data 
                                                     Train_RMSE   Test_RMSE
Alpha=0.064,Beta=0.053,Gamma=0.0,TripleExponent...   18.414568   20.990268
Alpha=0.3,Beta=0.4,Gamma=0.3,TripleExponentialS...   28.111886   10.945435
RegressionOnTime                                     30.718135   15.268955
Alpha=0.3,SimpleExponentialSmoothing                 32.470164   47.504821
Alpha=0.3,Beta=0.3,DoubleExponentialSmoothing        35.944983  265.567594
Simple Average                                       36.034234   53.460570
Alpha=0.995,SimpleExponentialSmoothing               38.714722   79.609847
Naive Model                                          45.063760   79.718773
2pointTrailingMovingAverage                                NaN   11.529278
4pointTrailingMovingAverage                                NaN   14.451403
6pointTrailingMovingAverage                                NaN   14.566327
9pointTrailingMovingAverage                                NaN   14.727630


from statsmodels.tsa.stattools import adfuller
def test_stationarity(timeseries):
    rolmean = timeseries.rolling(window=7).mean() #determining the rolling mean
    rolstd = timeseries.rolling(window=7).std() #determining the rolling standard deviation
    orig = plt.plot(timeseries, color='blue',label='Original')#Plot rolling statistics:
    mean = plt.plot(rolmean, color='red', label='Rolling Mean')
    std = plt.plot(rolstd, color='black', label = 'Rolling Std')
    plt.legend(loc='best')
    plt.title('Rolling Mean & Standard Deviation')
    plt.show(block=False)
    
    #Perform Dickey-Fuller test:
    print ('Results of Dickey-Fuller Test:')
    dftest = adfuller(timeseries, autolag='AIC')
    dfoutput = pd.Series(dftest[0:4], index=['Test Statistic','p-value','#Lags Used','Number of Observations Used'])
    
    for key,value in dftest[4].items():
        dfoutput['Critical Value (%s)'%key] = value
        print (dfoutput,'\n')


test_stationarity(df['Rose'])

Results of Dickey-Fuller Test:
Test Statistic                  -1.876699
p-value                          0.343101
#Lags Used                      13.000000
Number of Observations Used    173.000000
Critical Value (1%)             -3.468726
dtype: float64 

Test Statistic                  -1.876699
p-value                          0.343101
#Lags Used                      13.000000
Number of Observations Used    173.000000
Critical Value (1%)             -3.468726
Critical Value (5%)             -2.878396
dtype: float64 

Test Statistic                  -1.876699
p-value                          0.343101
#Lags Used                      13.000000
Number of Observations Used    173.000000
Critical Value (1%)             -3.468726
Critical Value (5%)             -2.878396
Critical Value (10%)            -2.575756
dtype: float64


# Let us take a difference of order 1 and check whether the Time Series is stationary or not.


test_stationarity(df['Rose'].diff().dropna())

Results of Dickey-Fuller Test:
Test Statistic                -8.044392e+00
p-value                        1.810895e-12
#Lags Used                     1.200000e+01
Number of Observations Used    1.730000e+02
Critical Value (1%)           -3.468726e+00
dtype: float64 

Test Statistic                -8.044392e+00
p-value                        1.810895e-12
#Lags Used                     1.200000e+01
Number of Observations Used    1.730000e+02
Critical Value (1%)           -3.468726e+00
Critical Value (5%)           -2.878396e+00
dtype: float64 

Test Statistic                -8.044392e+00
p-value                        1.810895e-12
#Lags Used                     1.200000e+01
Number of Observations Used    1.730000e+02
Critical Value (1%)           -3.468726e+00
Critical Value (5%)           -2.878396e+00
Critical Value (10%)          -2.575756e+00
dtype: float64


## The following loop helps us in getting a combination of different parameters of p and q in the range of 0 and 2
## We have kept the value of d as 1 as we need to take a difference of the series to make it stationary.
import itertools
p = q = range(0, 3)
d= range(1,2)
pdq = list(itertools.product(p, d, q))
print('Some parameter combinations for the Model...')
for i in range(1,len(pdq)):
    print('Model: {}'.format(pdq[i]))

Some parameter combinations for the Model...
Model: (0, 1, 1)
Model: (0, 1, 2)
Model: (1, 1, 0)
Model: (1, 1, 1)
Model: (1, 1, 2)
Model: (2, 1, 0)
Model: (2, 1, 1)
Model: (2, 1, 2)


# Creating an empty Dataframe with column names only
ARIMA_AIC = pd.DataFrame(columns=['param', 'AIC'])
ARIMA_AIC


from statsmodels.tsa.arima_model import ARIMA
import warnings
warnings.filterwarnings('ignore')
for param in pdq:
    ARIMA_model = ARIMA(train['Rose'].values,order=param).fit()
    print('ARIMA{} - AIC:{}'.format(param,ARIMA_model.aic))
    ARIMA_AIC = ARIMA_AIC.append({'param':param, 'AIC': ARIMA_model.aic}, ignore_index=True)

ARIMA(0, 1, 0) - AIC:1335.1526583086775
ARIMA(0, 1, 1) - AIC:1280.7261830464035
ARIMA(0, 1, 2) - AIC:1276.8353726229147
ARIMA(1, 1, 0) - AIC:1319.348310580781
ARIMA(1, 1, 1) - AIC:1277.77575701017
ARIMA(1, 1, 2) - AIC:1277.3592252057542
ARIMA(2, 1, 0) - AIC:1300.6092611745498
ARIMA(2, 1, 1) - AIC:1279.045689409309
ARIMA(2, 1, 2) - AIC:1279.2986939366306


## Sort the above AIC values in the ascending order to get the parameters for the minimum AIC value
ARIMA_AIC.sort_values(by='AIC',ascending=True)


auto_ARIMA = ARIMA(train['Rose'], order=(0,1,2),freq='M')
results_auto_ARIMA = auto_ARIMA.fit()
print(results_auto_ARIMA.summary())

                             ARIMA Model Results                              
==============================================================================
Dep. Variable:                 D.Rose   No. Observations:                  131
Model:                 ARIMA(0, 1, 2)   Log Likelihood                -634.418
Method:                       css-mle   S.D. of innovations             30.167
Date:                Thu, 28 Oct 2021   AIC                           1276.835
Time:                        18:25:20   BIC                           1288.336
Sample:                    02-29-1980   HQIC                          1281.509
                         - 12-31-1990                                         
================================================================================
                   coef    std err          z      P>|z|      [0.025      0.975]
--------------------------------------------------------------------------------
const           -0.4885      0.085     -5.742      0.000      -0.655      -0.322
ma.L1.D.Rose    -0.7601      0.101     -7.499      0.000      -0.959      -0.561
ma.L2.D.Rose    -0.2398      0.095     -2.518      0.012      -0.427      -0.053
                                    Roots                                    
=============================================================================
                  Real          Imaginary           Modulus         Frequency
-----------------------------------------------------------------------------
MA.1            1.0000           +0.0000j            1.0000            0.0000
MA.2           -4.1695           +0.0000j            4.1695            0.5000
-----------------------------------------------------------------------------


# Predict on the Test Set using this model and evaluate the model.


predicted_auto_ARIMA = results_auto_ARIMA.forecast(steps=len(test))


from sklearn.metrics import mean_squared_error
rmse = mean_squared_error(test['Rose'],predicted_auto_ARIMA[0],squared=False)
print(rmse)

15.61800957004907


results_DF= pd.DataFrame({'RMSE': [rmse]}
 ,index=['ARIMA(0,1,2)'])
results_DF


# Build an Automated version of a SARIMA model for which the best parameters are selected in accordance with the lowest Akaike Information Criteria (AIC)


import itertools
p = q = range(0, 3)
d= range(1,2)
D = range(0,1)
pdq = list(itertools.product(p, d, q))
model_pdq = [(x[0], x[1], x[2], 12) for x in list(itertools.product(p, D, q))]
print('Examples of some parameter combinations for Model...')
for i in range(1,len(pdq)):
    print('Model: {}{}'.format(pdq[i], model_pdq[i]))

Examples of some parameter combinations for Model...
Model: (0, 1, 1)(0, 0, 1, 12)
Model: (0, 1, 2)(0, 0, 2, 12)
Model: (1, 1, 0)(1, 0, 0, 12)
Model: (1, 1, 1)(1, 0, 1, 12)
Model: (1, 1, 2)(1, 0, 2, 12)
Model: (2, 1, 0)(2, 0, 0, 12)
Model: (2, 1, 1)(2, 0, 1, 12)
Model: (2, 1, 2)(2, 0, 2, 12)


SARIMA_AIC = pd.DataFrame(columns=['param','seasonal', 'AIC'])
SARIMA_AIC


import statsmodels.api as sm

for param in pdq:
    for param_seasonal in model_pdq:
        SARIMA_model = sm.tsa.statespace.SARIMAX(train['Rose'].values,
                                                 order=param,
                                                 seasonal_order=param_seasonal,
                                                 enforce_stationarity=False,
                                                 enforce_invertibility=False)
        results_SARIMA = SARIMA_model.fit(maxiter=1000)
        print('SARIMA{}x{} - AIC:{}'.format(param, param_seasonal, results_SARIMA.aic))
        SARIMA_AIC = SARIMA_AIC.append({'param':param,'seasonal':param_seasonal ,'AIC': results_SARIMA.aic}, ignore_index=True)

SARIMA(0, 1, 0)x(0, 0, 0, 12) - AIC:1323.9657875279158
SARIMA(0, 1, 0)x(0, 0, 1, 12) - AIC:1145.4230827207298
SARIMA(0, 1, 0)x(0, 0, 2, 12) - AIC:976.4375296380892
SARIMA(0, 1, 0)x(1, 0, 0, 12) - AIC:1139.921738995602
SARIMA(0, 1, 0)x(1, 0, 1, 12) - AIC:1116.020786938598
SARIMA(0, 1, 0)x(1, 0, 2, 12) - AIC:969.6913635752176
SARIMA(0, 1, 0)x(2, 0, 0, 12) - AIC:960.8812220353041
SARIMA(0, 1, 0)x(2, 0, 1, 12) - AIC:962.8794540697587
SARIMA(0, 1, 0)x(2, 0, 2, 12) - AIC:955.5735408945745
SARIMA(0, 1, 1)x(0, 0, 0, 12) - AIC:1263.5369097383968
SARIMA(0, 1, 1)x(0, 0, 1, 12) - AIC:1098.5554825918337
SARIMA(0, 1, 1)x(0, 0, 2, 12) - AIC:923.6314049383909
SARIMA(0, 1, 1)x(1, 0, 0, 12) - AIC:1095.793632491829
SARIMA(0, 1, 1)x(1, 0, 1, 12) - AIC:1054.7434330945437
SARIMA(0, 1, 1)x(1, 0, 2, 12) - AIC:918.857348330446
SARIMA(0, 1, 1)x(2, 0, 0, 12) - AIC:914.5982866535896
SARIMA(0, 1, 1)x(2, 0, 1, 12) - AIC:915.333243046167
SARIMA(0, 1, 1)x(2, 0, 2, 12) - AIC:901.1988260392989
SARIMA(0, 1, 2)x(0, 0, 0, 12) - AIC:1251.6675430541045
SARIMA(0, 1, 2)x(0, 0, 1, 12) - AIC:1083.4866975264888
SARIMA(0, 1, 2)x(0, 0, 2, 12) - AIC:913.4938486617702
SARIMA(0, 1, 2)x(1, 0, 0, 12) - AIC:1088.8332843414148
SARIMA(0, 1, 2)x(1, 0, 1, 12) - AIC:1045.5400933520334
SARIMA(0, 1, 2)x(1, 0, 2, 12) - AIC:904.8310913572031
SARIMA(0, 1, 2)x(2, 0, 0, 12) - AIC:913.0105912257992
SARIMA(0, 1, 2)x(2, 0, 1, 12) - AIC:914.1707545032118
SARIMA(0, 1, 2)x(2, 0, 2, 12) - AIC:887.9375085679931
SARIMA(1, 1, 0)x(0, 0, 0, 12) - AIC:1308.161871082466
SARIMA(1, 1, 0)x(0, 0, 1, 12) - AIC:1135.2955447585703
SARIMA(1, 1, 0)x(0, 0, 2, 12) - AIC:963.9405391257686
SARIMA(1, 1, 0)x(1, 0, 0, 12) - AIC:1124.8860786804528
SARIMA(1, 1, 0)x(1, 0, 1, 12) - AIC:1105.4080055021445
SARIMA(1, 1, 0)x(1, 0, 2, 12) - AIC:958.5001972948426
SARIMA(1, 1, 0)x(2, 0, 0, 12) - AIC:939.0984778664011
SARIMA(1, 1, 0)x(2, 0, 1, 12) - AIC:940.908713366107
SARIMA(1, 1, 0)x(2, 0, 2, 12) - AIC:942.2973103070892
SARIMA(1, 1, 1)x(0, 0, 0, 12) - AIC:1262.1840064255507
SARIMA(1, 1, 1)x(0, 0, 1, 12) - AIC:1094.317270864078
SARIMA(1, 1, 1)x(0, 0, 2, 12) - AIC:923.0862224063895
SARIMA(1, 1, 1)x(1, 0, 0, 12) - AIC:1083.3937965030339
SARIMA(1, 1, 1)x(1, 0, 1, 12) - AIC:1054.7180547135608
SARIMA(1, 1, 1)x(1, 0, 2, 12) - AIC:916.3549428498078
SARIMA(1, 1, 1)x(2, 0, 0, 12) - AIC:905.9249060842012
SARIMA(1, 1, 1)x(2, 0, 1, 12) - AIC:907.2972867470766
SARIMA(1, 1, 1)x(2, 0, 2, 12) - AIC:900.6725795936867
SARIMA(1, 1, 2)x(0, 0, 0, 12) - AIC:1251.949504070612
SARIMA(1, 1, 2)x(0, 0, 1, 12) - AIC:1085.486192810111
SARIMA(1, 1, 2)x(0, 0, 2, 12) - AIC:915.4938402457078
SARIMA(1, 1, 2)x(1, 0, 0, 12) - AIC:1090.7760922916361
SARIMA(1, 1, 2)x(1, 0, 1, 12) - AIC:1042.6183211002572
SARIMA(1, 1, 2)x(1, 0, 2, 12) - AIC:906.7318500262634
SARIMA(1, 1, 2)x(2, 0, 0, 12) - AIC:906.1690199910133
SARIMA(1, 1, 2)x(2, 0, 1, 12) - AIC:907.4597828543872
SARIMA(1, 1, 2)x(2, 0, 2, 12) - AIC:889.871766869675
SARIMA(2, 1, 0)x(0, 0, 0, 12) - AIC:1280.2537561535767
SARIMA(2, 1, 0)x(0, 0, 1, 12) - AIC:1128.7773704710878
SARIMA(2, 1, 0)x(0, 0, 2, 12) - AIC:958.0793208829955
SARIMA(2, 1, 0)x(1, 0, 0, 12) - AIC:1099.5086021575923
SARIMA(2, 1, 0)x(1, 0, 1, 12) - AIC:1076.7863198641112
SARIMA(2, 1, 0)x(1, 0, 2, 12) - AIC:951.1988165559382
SARIMA(2, 1, 0)x(2, 0, 0, 12) - AIC:924.6004792645502
SARIMA(2, 1, 0)x(2, 0, 1, 12) - AIC:925.9757801383555
SARIMA(2, 1, 0)x(2, 0, 2, 12) - AIC:927.8380693280792
SARIMA(2, 1, 1)x(0, 0, 0, 12) - AIC:1263.2315231797147
SARIMA(2, 1, 1)x(0, 0, 1, 12) - AIC:1094.2093491949397
SARIMA(2, 1, 1)x(0, 0, 2, 12) - AIC:922.9408472121463
SARIMA(2, 1, 1)x(1, 0, 0, 12) - AIC:1071.4249601101183
SARIMA(2, 1, 1)x(1, 0, 1, 12) - AIC:1052.924447120346
SARIMA(2, 1, 1)x(1, 0, 2, 12) - AIC:916.2424912822876
SARIMA(2, 1, 1)x(2, 0, 0, 12) - AIC:896.5181608176032
SARIMA(2, 1, 1)x(2, 0, 1, 12) - AIC:897.6399565369805
SARIMA(2, 1, 1)x(2, 0, 2, 12) - AIC:899.4835866298606
SARIMA(2, 1, 2)x(0, 0, 0, 12) - AIC:1253.9102116146694
SARIMA(2, 1, 2)x(0, 0, 1, 12) - AIC:1085.9643552598197
SARIMA(2, 1, 2)x(0, 0, 2, 12) - AIC:916.3258311100999
SARIMA(2, 1, 2)x(1, 0, 0, 12) - AIC:1073.291271370496
SARIMA(2, 1, 2)x(1, 0, 1, 12) - AIC:1044.1909353602505
SARIMA(2, 1, 2)x(1, 0, 2, 12) - AIC:907.6661488808384
SARIMA(2, 1, 2)x(2, 0, 0, 12) - AIC:897.3464442600842
SARIMA(2, 1, 2)x(2, 0, 1, 12) - AIC:898.3781891066726
SARIMA(2, 1, 2)x(2, 0, 2, 12) - AIC:890.6687985705821


SARIMA_AIC.sort_values(by=['AIC']).head()


import statsmodels.api as sm

auto_SARIMA_12 = sm.tsa.statespace.SARIMAX(train['Rose'].values,
                                         order=(0, 1, 2),
                                         seasonal_order=(2, 0, 2, 12),
                                         enforce_stationarity=False,
                                         enforce_invertibility=False)
 
results_auto_SARIMA_12 = auto_SARIMA_12.fit(maxiter=1000)
print(results_auto_SARIMA_12.summary())

                                     SARIMAX Results                                      
==========================================================================================
Dep. Variable:                                  y   No. Observations:                  132
Model:             SARIMAX(0, 1, 2)x(2, 0, 2, 12)   Log Likelihood                -436.969
Date:                            Thu, 28 Oct 2021   AIC                            887.938
Time:                                    18:31:16   BIC                            906.448
Sample:                                         0   HQIC                           895.437
                                            - 132                                         
Covariance Type:                              opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
ma.L1         -0.8427    189.863     -0.004      0.996    -372.968     371.282
ma.L2         -0.1573     29.829     -0.005      0.996     -58.620      58.306
ar.S.L12       0.3467      0.079      4.375      0.000       0.191       0.502
ar.S.L24       0.3023      0.076      3.996      0.000       0.154       0.451
ma.S.L12       0.0767      0.133      0.577      0.564      -0.184       0.337
ma.S.L24      -0.0726      0.146     -0.498      0.618      -0.358       0.213
sigma2       251.3137   4.77e+04      0.005      0.996   -9.33e+04    9.38e+04
===================================================================================
Ljung-Box (L1) (Q):                   0.10   Jarque-Bera (JB):                 2.33
Prob(Q):                              0.75   Prob(JB):                         0.31
Heteroskedasticity (H):               0.88   Skew:                             0.37
Prob(H) (two-sided):                  0.70   Kurtosis:                         3.03
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).


results_auto_SARIMA_12.plot_diagnostics()
plt.show()


# Predict on the Test Set using this model and evaluate the model.


predicted_auto_SARIMA_12 = results_auto_SARIMA_12.get_forecast(steps=len(test))


predicted_auto_SARIMA_12.summary_frame(alpha=0.05).head()


rmse = mean_squared_error(test['Rose'],predicted_auto_SARIMA_12.predicted_mean,squared=False)
print(rmse)

26.928361850623816


temp_resultsDf = pd.DataFrame({'RMSE': [rmse]}
 ,index=['SARIMA(0,1,2)(2,0,2,12)'])
results_DF = pd.concat([results_DF,temp_resultsDf])
results_DF


from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
plot_acf(df['Rose'].diff().dropna(),lags=50,title='Differenced Data Autocorrelation')
plot_pacf(df['Rose'].diff().dropna(),lags=50,title='Differenced Data Patial Autocorrelation')
plt.show()


df.plot()
plt.grid();


(df['Rose'].diff(12)).plot()
plt.grid();


(df['Rose'].diff(12)).diff().plot()
plt.grid();


# Let us go ahead and check the stationarity of the above series before fitting the SARIMA model.
test_stationarity((train['Rose'].diff(12).dropna()).diff(1).dropna())

Results of Dickey-Fuller Test:
Test Statistic                  -3.692348
p-value                          0.004222
#Lags Used                      11.000000
Number of Observations Used    107.000000
Critical Value (1%)             -3.492996
dtype: float64 

Test Statistic                  -3.692348
p-value                          0.004222
#Lags Used                      11.000000
Number of Observations Used    107.000000
Critical Value (1%)             -3.492996
Critical Value (5%)             -2.888955
dtype: float64 

Test Statistic                  -3.692348
p-value                          0.004222
#Lags Used                      11.000000
Number of Observations Used    107.000000
Critical Value (1%)             -3.492996
Critical Value (5%)             -2.888955
Critical Value (10%)            -2.581393
dtype: float64


# Checking the ACF and the PACF plots for the new modified Time Series
plot_acf((df['Rose'].diff(12).dropna()).diff(1).dropna(),lags=30)
plot_pacf((df['Rose'].diff(12).dropna()).diff(1).dropna(),lags=30);


# We are going to take the seasonal period as 12.


manual_SARIMA_12= sm.tsa.statespace.SARIMAX(train['Rose'].values,
 order=(2, 1, 2),
 seasonal_order=(2, 0, 2, 12),
 enforce_stationarity=False,
 enforce_invertibility=False)
results_manual_SARIMA_12 = manual_SARIMA_12.fit(maxiter=1000)
print(results_manual_SARIMA_12.summary())

                                     SARIMAX Results                                      
==========================================================================================
Dep. Variable:                                  y   No. Observations:                  132
Model:             SARIMAX(2, 1, 2)x(2, 0, 2, 12)   Log Likelihood                -436.334
Date:                            Thu, 28 Oct 2021   AIC                            890.669
Time:                                    18:35:41   BIC                            914.468
Sample:                                         0   HQIC                           900.311
                                            - 132                                         
Covariance Type:                              opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
ar.L1          0.3972      0.322      1.232      0.218      -0.234       1.029
ar.L2         -0.1377      0.115     -1.202      0.229      -0.362       0.087
ma.L1         -1.2536    223.406     -0.006      0.996    -439.121     436.614
ma.L2          0.2536     56.688      0.004      0.996    -110.853     111.360
ar.S.L12       0.3403      0.078      4.342      0.000       0.187       0.494
ar.S.L24       0.2965      0.071      4.156      0.000       0.157       0.436
ma.S.L12       0.1054      0.131      0.805      0.421      -0.151       0.362
ma.S.L24      -0.0619      0.153     -0.405      0.686      -0.362       0.238
sigma2       247.1698   5.52e+04      0.004      0.996   -1.08e+05    1.08e+05
===================================================================================
Ljung-Box (L1) (Q):                   0.05   Jarque-Bera (JB):                 2.58
Prob(Q):                              0.83   Prob(JB):                         0.27
Heteroskedasticity (H):               0.95   Skew:                             0.37
Prob(H) (two-sided):                  0.87   Kurtosis:                         3.23
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).


results_manual_SARIMA_12.plot_diagnostics()
plt.show()


# Predict on the Test Set using this model and evaluate the model.


predicted_manual_SARIMA_12 = results_manual_SARIMA_12.get_forecast(steps=len(test))


predicted_manual_SARIMA_12.summary_frame(alpha=0.05).head()


# RMSE Score
rmse = mean_squared_error(test['Rose'],predicted_manual_SARIMA_12.predicted_mean,squared=False)
print(rmse)

27.463437070869865


temp_resultsDf = pd.DataFrame({'RMSE': [rmse]}
 ,index=['SARIMA(2,1,2)(2,0,2,12)'])
results_DF = pd.concat([results_DF,temp_resultsDf])
results_DF


# Manual ARIMA model
###Let us look once agin ACF and PACF plots
plot_acf(df['Rose'].diff().dropna(),lags=50,title='Differenced Data Autocorrelation')
plot_pacf(df['Rose'].diff().dropna(),lags=50,title='Differenced Data Partial Autocorrelation')
plt.show()


manual_ARIMA = ARIMA(train['Rose'].astype('float64'), order=(2,1,2),freq='M')
results_manual_ARIMA = manual_ARIMA.fit()
print(results_manual_ARIMA.summary())

                             ARIMA Model Results                              
==============================================================================
Dep. Variable:                 D.Rose   No. Observations:                  131
Model:                 ARIMA(2, 1, 2)   Log Likelihood                -633.649
Method:                       css-mle   S.D. of innovations             29.975
Date:                Thu, 28 Oct 2021   AIC                           1279.299
Time:                        18:37:32   BIC                           1296.550
Sample:                    02-29-1980   HQIC                          1286.309
                         - 12-31-1990                                         
================================================================================
                   coef    std err          z      P>|z|      [0.025      0.975]
--------------------------------------------------------------------------------
const           -0.4911      0.081     -6.076      0.000      -0.649      -0.333
ar.L1.D.Rose    -0.4383      0.218     -2.015      0.044      -0.865      -0.012
ar.L2.D.Rose     0.0269      0.109      0.246      0.806      -0.188       0.241
ma.L1.D.Rose    -0.3316      0.203     -1.633      0.102      -0.729       0.066
ma.L2.D.Rose    -0.6684      0.201     -3.332      0.001      -1.062      -0.275
                                    Roots                                    
=============================================================================
                  Real          Imaginary           Modulus         Frequency
-----------------------------------------------------------------------------
AR.1           -2.0290           +0.0000j            2.0290            0.5000
AR.2           18.3391           +0.0000j           18.3391            0.0000
MA.1            1.0000           +0.0000j            1.0000            0.0000
MA.2           -1.4961           +0.0000j            1.4961            0.5000
-----------------------------------------------------------------------------


# Predict on the Test Set using this model and evaluate the model
predicted_manual_ARIMA = results_manual_ARIMA.forecast(steps=len(test))


rmse = mean_squared_error(test['Rose'],predicted_manual_ARIMA[0],squared=False)
print(rmse)

15.354883102386067


temp_resultsDf = pd.DataFrame({'RMSE': [rmse]}
 ,index=['ARIMA(2,1,2)'])
results_DF = pd.concat([results_DF,temp_resultsDf])
results_DF


results_DF = results_DF.rename(columns={'RMSE':'Test_RMSE'})


results_DF


Final_df1=result_DF.copy()
Final_df1


Final_df1.drop('Train_RMSE',axis=1,inplace=True)


Final_df1


Final_Model_results=pd.concat([Final_df1,results_DF])
Final_Model_results


# Final Model_Results of all The models sorted by RMSE score for Test set


Final_Model_results.sort_values(by=['Test_RMSE'],ascending=True)


TES_Final=df.copy()


model_TES=ExponentialSmoothing(TES_Final['Rose'],trend='additive',seasonal='multiplicative',freq='M')


model_TES_autofit = model_TES.fit()


model_TES_autofit.params

{'smoothing_level': 0.1068894315723457,
 'smoothing_trend': 8.824151305370089e-07,
 'smoothing_seasonal': 0.0002559971640309986,
 'damping_trend': nan,
 'initial_level': 44.696998132215775,
 'initial_trend': -0.16969823845873067,
 'initial_seasons': array([2.41432051, 2.74182722, 3.02430727, 2.70264271, 2.95424019,
        3.1996946 , 3.58978358, 3.69945974, 3.49289704, 3.45210178,
        4.00705632, 5.52392935]),
 'use_boxcox': False,
 'lamda': None,
 'remove_bias': False}


TES_Final['auto']=model_TES_autofit.fittedvalues
TES_Final


TES_Future= model_TES_autofit.forecast(steps=12)
TES_Future

1995-08-31    47.543980
1995-09-30    44.305549
1995-10-31    43.204999
1995-11-30    49.471255
1995-12-31    67.258354
1996-01-31    28.980369
1996-02-29    32.453941
1996-03-31    35.286056
1996-04-30    31.082453
1996-05-31    33.460913
1996-06-30    35.698092
1996-07-31    39.441978
Freq: M, dtype: float64


# Plotting The Forecast with Confidence Bond
plt.figure(figsize=(18,9))
plt.plot(TES_Final,label='Current')
plt.plot(TES_Future,label='Future')
plt.plot(TES_Future, label=['Alpha=0.107,Beta=1.38,Gamma=0.0002,TripleExponentialSmoothing prediction in future for 12 Months'])
plt.legend(loc='best')
plt.grid();


# Final Result
rmse= metrics.mean_squared_error(TES_Final['Rose'],TES_Final['auto'],squared=False)
print('For Alpha=0.107,Beta=1.38,Gamma=0.0002 , Triple exponential Smoothing model Forecasting',rmse)

For Alpha=0.107,Beta=1.38,Gamma=0.0002 , Triple exponential Smoothing model Forecasting 16.09622690763845


# Predicted Model for the Sparkling Sales for final Model
TES_Final['Rose'].plot()
TES_Future.plot()

<AxesSubplot:xlabel='Time_Stamp'>


TES_Final_1=pd.DataFrame({'lower_CI':TES_Future - ((np.percentile(TES_Future,[2.5,97.5])))[0],
                         'Forecasting': TES_Future,
                         'upper_CI': TES_Future + ((np.percentile(TES_Future,[2.5,97.5])))[1]})


axis=df['Rose'].plot(label='Actual',figsize=(15,8))
TES_Final_1.plot(ax=axis,label='Forecast',alpha=0.5)
axis.fill_between(TES_Final_1.index, TES_Final_1['lower_CI'],TES_Final_1['upper_CI'],color='k',alpha=.15)
axis.set_xlabel('Year-Month')
axis.set_ylabel('Rose')
plt.legend(loc='best')
plt.show();

	Rose
count	187.000000
mean	89.914439
std	39.238325
min	28.000000
25%	62.500000
50%	85.000000
75%	111.000000
max	267.000000

Time_Stamp	April	August	December	February	January	July	June	March	May	November	October	September
Time_Stamp
1980	99.0	129.000000	267.0	118.0	112.0	118.000000	168.0	129.0	116.0	150.0	147.0	205.0
1981	97.0	214.000000	226.0	129.0	126.0	222.000000	127.0	124.0	102.0	154.0	141.0	118.0
1982	97.0	117.000000	169.0	77.0	89.0	117.000000	121.0	82.0	127.0	134.0	112.0	106.0
1983	85.0	124.000000	164.0	108.0	75.0	109.000000	108.0	115.0	101.0	135.0	95.0	105.0
1984	87.0	142.000000	159.0	85.0	88.0	87.000000	87.0	112.0	91.0	139.0	108.0	95.0
1985	93.0	103.000000	129.0	82.0	61.0	87.000000	75.0	124.0	108.0	123.0	108.0	90.0
1986	71.0	118.000000	141.0	65.0	57.0	110.000000	67.0	67.0	76.0	107.0	85.0	99.0
1987	86.0	73.000000	157.0	65.0	58.0	87.000000	74.0	70.0	93.0	96.0	100.0	101.0
1988	66.0	77.000000	135.0	115.0	63.0	79.000000	83.0	70.0	67.0	100.0	116.0	102.0
1989	74.0	74.000000	137.0	60.0	71.0	86.000000	91.0	89.0	73.0	109.0	87.0	87.0
1990	77.0	70.000000	132.0	69.0	43.0	78.000000	76.0	73.0	69.0	110.0	65.0	83.0
1991	65.0	55.000000	106.0	55.0	54.0	96.000000	65.0	66.0	60.0	74.0	63.0	71.0
1992	53.0	52.000000	91.0	47.0	34.0	67.000000	55.0	56.0	53.0	58.0	51.0	46.0
1993	45.0	54.000000	77.0	40.0	33.0	57.000000	55.0	46.0	41.0	48.0	52.0	46.0
1994	48.0	45.666667	84.0	35.0	30.0	45.333333	45.0	42.0	44.0	63.0	51.0	46.0
1995	52.0	NaN	NaN	39.0	30.0	62.000000	40.0	45.0	28.0	NaN	NaN	NaN

y	mean	mean_se	mean_ci_lower	mean_ci_upper
0	62.867264	15.928501	31.647976	94.086552
1	70.541190	16.147659	38.892361	102.190020
2	77.356411	16.147656	45.707586	109.005235
3	76.208814	16.147656	44.559989	107.857639
4	72.747398	16.147656	41.098573	104.396223

y	mean	mean_se	mean_ci_lower	mean_ci_upper
0	62.232440	15.796450	31.271968	93.192913
1	69.150517	15.979715	37.830851	100.470182
2	76.828021	16.016818	45.435636	108.220407
3	76.627751	16.029605	45.210302	108.045201
4	73.150494	16.028896	41.734436	104.566552

Q1. Read the data as an appropriate Time Series data and plot the data.¶

Q2. Perform appropriate Exploratory Data Analysis to understand the data and also perform decomposition.¶

Q3.Split the data into training and test. The test data should start in 1991.¶

Q4). Build various exponential smoothing models on the training data and evaluate the model usingRMSE on the test data. Other models such as regression,naïve forecast models and simple averagemodels. should also be built on the training data and check the performance on the test data using RMSE¶

6. Build an automated version of the ARIMA/SARIMA model in which the parameters are selected using the lowest Akaike Information Criteria (AIC) on the training data and evaluate this model on the test data using RMSE.¶

7. Build ARIMA/SARIMA models based on the cut-off points of ACF and PACF on the training data and evaluate this model on the test data using RMSE.¶

8. Build a table (create a data frame) with all the models built along with their corresponding parameters and the respective RMSE values on the test data.¶

9. Based on the model-building exercise, build the most optimum model(s) on the complete data and predict 12 months into the future with appropriate confidence intervals/bands.¶

	YearMonth	Rose
0	1980-01	112.0
1	1980-02	118.0
2	1980-03	129.0
3	1980-04	99.0
4	1980-05	116.0

	YearMonth	Rose	Time_Stamp
0	1980-01	112.0	1980-01-31
1	1980-02	118.0	1980-02-29
2	1980-03	129.0	1980-03-31
3	1980-04	99.0	1980-04-30
4	1980-05	116.0	1980-05-31

	Rose
Time_Stamp
1990-08-31	70.0
1990-09-30	83.0
1990-10-31	65.0
1990-11-30	110.0
1990-12-31	132.0

	Rose
Time_Stamp
1991-01-31	54.0
1991-02-28	55.0
1991-03-31	66.0
1991-04-30	65.0
1991-05-31	60.0

	Rose
Time_Stamp
1995-03-31	45.0
1995-04-30	52.0
1995-05-31	28.0
1995-06-30	40.0
1995-07-31	62.0

	Train_RMSE	Test_RMSE
RegressionOnTime	30.718135	15.268955
Naive Model	45.063760	79.718773

	Alpha_Values	Train_RMSE	Test_RMSE
0	0.3	32.470164	47.504821
1	0.4	33.035130	53.767406
2	0.5	33.682839	59.641786
3	0.6	34.441171	64.971288
4	0.7	35.323261	69.698162
5	0.8	36.334596	73.773992
6	0.9	37.482782	77.139276

	Alpha_Values	Beta_Values	Train_RMSE	Test_RMSE
0	0.3	0.3	35.944983	265.567594
1	0.3	0.4	37.393239	358.750942
2	0.3	0.5	38.888325	451.810230
3	0.3	0.6	40.280670	536.954148
4	0.3	0.7	41.403518	601.616351
...	...	...	...	...
59	1.0	0.6	51.831610	801.680218
60	1.0	0.7	54.497039	841.892573
61	1.0	0.8	57.365879	853.965537
62	1.0	0.9	60.474309	834.710935
63	1.0	1.0	63.873454	780.079579

	Alpha_Values	Beta_Values	Train_RMSE	Test_RMSE
0	0.3	0.3	35.944983	265.567594
8	0.4	0.3	36.749123	339.306534
1	0.3	0.4	37.393239	358.750942
16	0.5	0.3	37.433314	394.272629
24	0.6	0.3	38.348984	439.296033

	Alpha_Values	Beta_Values	Gamma_Values	Train_RMSE	Test_RMSE
0	0.3	0.3	0.3	27.217969	19.057218
1	0.3	0.3	0.4	27.399095	11.201633
2	0.3	0.3	0.5	27.928512	30.565763
3	0.3	0.3	0.6	28.888611	63.623019
4	0.3	0.3	0.7	30.568635	122.472557
...	...	...	...	...	...
507	1.0	1.0	0.6	28358.458519	9603.635095
508	1.0	1.0	0.7	30724.126331	23029.955361
509	1.0	1.0	0.8	1218.755446	9626.710854
510	1.0	1.0	0.9	14150.253251	9691.905402
511	1.0	1.0	1.0	1768.254189	8138.618579

	Alpha_Values	Beta_Values	Gamma_Values	Train_RMSE	Test_RMSE
8	0.3	0.4	0.3	28.111886	10.945435
1	0.3	0.3	0.4	27.399095	11.201633
69	0.4	0.3	0.8	32.601491	12.615607
16	0.3	0.5	0.3	29.087520	14.414604
131	0.5	0.3	0.6	32.144773	16.720720

	param	AIC
2	(0, 1, 2)	1276.835373
5	(1, 1, 2)	1277.359225
4	(1, 1, 1)	1277.775757
7	(2, 1, 1)	1279.045689
8	(2, 1, 2)	1279.298694
1	(0, 1, 1)	1280.726183
6	(2, 1, 0)	1300.609261
3	(1, 1, 0)	1319.348311
0	(0, 1, 0)	1335.152658

	param	seasonal	AIC
26	(0, 1, 2)	(2, 0, 2, 12)	887.937509
53	(1, 1, 2)	(2, 0, 2, 12)	889.871767
80	(2, 1, 2)	(2, 0, 2, 12)	890.668799
69	(2, 1, 1)	(2, 0, 0, 12)	896.518161
78	(2, 1, 2)	(2, 0, 0, 12)	897.346444

	RMSE
ARIMA(0,1,2)	15.618010
SARIMA(0,1,2)(2,0,2,12)	26.928362
SARIMA(2,1,2)(2,0,2,12)	27.463437

	Test_RMSE
Alpha=0.3,Beta=0.4,Gamma=0.3,TripleExponentialSmoothing	10.945435
2pointTrailingMovingAverage	11.529278
4pointTrailingMovingAverage	14.451403
6pointTrailingMovingAverage	14.566327
9pointTrailingMovingAverage	14.727630
RegressionOnTime	15.268955
ARIMA(2,1,2)	15.354883
ARIMA(0,1,2)	15.618010
Alpha=0.064,Beta=0.053,Gamma=0.0,TripleExponentialSmoothing	20.990268
SARIMA(0,1,2)(2,0,2,12)	26.928362
SARIMA(2,1,2)(2,0,2,12)	27.463437
Alpha=0.3,SimpleExponentialSmoothing	47.504821
Simple Average	53.460570
Alpha=0.995,SimpleExponentialSmoothing	79.609847
Naive Model	79.718773
Alpha=0.3,Beta=0.3,DoubleExponentialSmoothing	265.567594